1. Races
- "A beats B by x m" in a race of D m: when A finishes D, B has run D − x.
- "A gives B a start of x m": B starts x m ahead; A runs D, B runs D − x.
- "A beats B by t seconds": B takes t s more; B covers x m in t s.
- Speeds are in the ratio of distances covered in the same time.
In a 100 m race A beats B by 10 m and B beats C by 10 m. When A runs 100, B runs 90, C runs 90 × 0.9 = 81 → A beats C by 19 m.
A runs 1 km in 3 min, B in 3 min 20 s → A beats B by 20 s; B's speed = 1000/200 = 5 m/s → by 100 m.
2. Circular track
- Same direction: meet every L/(a − b); opposite: L/(a + b).
- All meet at the start again after LCM of their lap times.
Lap times 4, 6 and 10 min → together at start after LCM = 60 min.
3. Clocks — the basics
- Minute hand: 6° per minute. Hour hand: 0.5° per minute (30° per hour).
- Relative speed = 5.5° per minute; minute hand gains 55 minute-spaces in 60 minutes.
- Angle at H:M = |30H − 5.5M| (take 360 − result if over 180).
- Hands coincide 11 times in 12 hours (22 in a day), every 65 5/11 min. They are opposite or at right angles: 22 and 44 times a day.
Angle at 3:40 = |90 − 220| = 130°.
Between 4 and 5, hands together at (20 × 12/11) = 21 9/11 min past 4.
4. Fast or slow clocks
A clock loses 5 min every hour. Set right at 6 a.m.; true time when it shows 6 p.m.: it moves 55 min per 60 → 12 h shown = 12 × 60/55 = 13 1/11 h → about 7:05 p.m.
5. Calendar quick facts
- Odd days: normal year 1, leap year 2. 100 years = 5, 200 = 3, 300 = 1, 400 = 0.
- A leap year is divisible by 4 (century years by 400).
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
ASO exam level
Example 1. In a 1 km race, A beats B by 28 metres or 5 seconds. A's time over the course is:
(a) 183.57 s (b) 173.57 s (c) 168.57 s (d) 178.57 s
Answer: (b) 173.57 s
Solution: B runs 28 m in 5 s → 5.6 m/s. B's time for 1000 m = 178.57 s. A is 5 s faster: 173.57 s.
Example 2. A runs 5/4 times as fast as B. In a 100 m race, how much start should A give B so that the race ends in a dead heat?
(a) 20 m (b) 25 m (c) 30 m (d) 23 m
Answer: (a) 20 m
Solution: When A runs 100 m, B runs 100 × 4/5 = 80 m. Start = 100 − 80 = 20 m.
Advanced level
Example 3. In a 100 m race, A runs at 9 km/h. A gives B a start of 5 m and still beats him by 15 seconds. B's speed is:
(a) 7.22 km/h (b) 6.22 km/h (c) 5.72 km/h (d) 6.55 km/h
Answer: (b) 6.22 km/h
Solution: A's time = 100 ÷ (9×5/18) = 40 s. B takes 40 + 15 = 55 s for 95 m → 1.73 m/s = 6.22 km/h.
Example 4. In a 100 m race, A runs at 10 km/h. A gives B a start of 4 m and still beats him by 15 seconds. B's speed is:
(a) 6.78 km/h (b) 7.78 km/h (c) 6.28 km/h (d) 7.06 km/h
Answer: (a) 6.78 km/h
Solution: A's time = 100 ÷ (10×5/18) = 36 s. B takes 36 + 15 = 51 s for 96 m → 1.88 m/s = 6.78 km/h.
Example 5. A and B start together from the same point and run around a circular track of 1200 m at 6 m/s and 4 m/s in opposite directions. When will they meet for the first time?
(a) 200 s (b) 120 s (c) 240 s (d) 600 s
Answer: (b) 120 s
Solution: Opposite directions: relative speed = 6 + 4 = 10 m/s. First meeting = 1200/10 = 120 s. (Same direction would be 1200/2 = 600 s.)
Example 6. A and B start together from the same point and run around a circular track of 1200 m at 6 m/s and 4 m/s in opposite directions. When will they meet for the first time?
(a) 600 s (b) 200 s (c) 120 s (d) 240 s
Answer: (c) 120 s
Solution: Opposite directions: relative speed = 6 + 4 = 10 m/s. First meeting = 1200/10 = 120 s. (Same direction would be 1200/2 = 600 s.)
Example 7. A clock is set right at 5 AM. It loses 16 minutes in 24 hours. What is the true time when the clock shows 10 PM on the 4th day?
(a) 11:15 PM (b) 9:44 PM (c) 10:00 PM (d) 11:00 PM
Answer: (d) 11:00 PM
Solution: Clock time passed = 89 h. Clock shows 1424 min for every 1440 true min. True time passed = 89 × 1440/1424 h → 11:00 PM.
Topic-wise practice for Races and clocks (Quant): Races, circular tracks, angle between clock hands. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- A beats B by x metres or seconds
- Dead heat and head start
- Circular track meeting time
- Angle between hands at a given time
- When the hands coincide, are opposite or at right angles
- Clock gaining or losing time